Comparing Multiagent Systems Research inCombinatorial Auctions and Voting∗

Vincent ConitzerDepartment of Computer Science

Duke UniversityDurham, NC 27708, USA

conitzer@cs.duke.edu

Abstract

In a combinatorial auction, a set of items is for sale, and agents can bid onsubsets of these items. In a voting setting, the agents decide among a set of alter-natives by having each agent rank all the alternatives. Many of the key researchissues in these two domains are similar. The aim of this paper is to give a conve-nient side-by-side comparison that will clarify the relation between the domains,and serve as a guide to future research.

1 IntroductionIn multiagent systems, it is often necessary for a group of agents to make a collectivedecision even though they have different preferences over the different options (a de-tailed discussion can be found in the article [31]). For example, the agents may haveto decide how to allocate a set of items among themselves. A common mechanism fordoing this is to run a combinatorial auction, where the agents place bids on bundles(subsets) of items (for example, an agent can bid 10 on the bundle consisting of itemsa and c), and based on these bids an allocation of the items is determined, as well asthe payment that each agent needs to make. However, there are many other collectivedecision problems that do not involve allocating items or making payments. A generalapproach for choosing among a set of alternatives is for each agent to rank all the al-ternatives, after which a winning alternative is chosen based on these rankings. In thiscase, we say that the agents vote over the alternatives (and the rankings are the votes).

Combinatorial auctions have become a well-established research topic in multi-agent systems. In recent years, research on voting in multiagent systems has alsosoared, and a community of researchers interested in computational social choice has

∗This paper corresponds to an invited talk in the session on Computation and Social Choice at the TenthInternational Symposium on Artificial Intelligence and Mathematics (ISAIM-08). An early version of thispaper appeared in the informal electronic proceedings of ISAIM-08. The author is supported by an AlfredP. Sloan Research Fellowship and by NSF under award numbers IIS-0812113 and CAREER-0953756.

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formed. While there is certainly significant interaction between the researchers work-ing on combinatorial auctions and the ones working on voting, the two communitiesare perhaps more disjoint than one would expect. This paper aims to compare some ofthe research issues across the two topics, thereby providing to each community a con-venient window into the other (and, perhaps, a window into both for outsiders). Thecomparison is also likely to suggest new research directions for each topic. Finally, weargue that the research on the two topics is likely to converge further in the future.

2 Defining the rulesBoth in combinatorial auctions and in voting settings, we need to specify how agentsreport their preferences (that is, how they place their bids/cast their votes), and how anoutcome is chosen on the basis of these. As we will see, there are multiple ways ofdoing so in each case.

2.1 Combinatorial auctionsTo understand the different ways in which a combinatorial auction can be designed, itis helpful to first study some common single-item auctions.

• English auction. In an English auction (perhaps the best-known auction format)any bidder can enter a bid higher than the current highest bid at any point. Oncenobody wants to submit a higher bid, the current highest bidder wins the itemand pays her bid.

• Japanese auction. In a Japanese auction, there is an initial price of zero for theitem, which is gradually increased. A bidder can leave the room at any pointif the price becomes too high for her. The auction ends when only one bidderremains, who then wins and pays the final price.

• Dutch auction. In a Dutch auction, the price starts at a high value and is graduallydecreased. At any point, any bidder can claim the item at the current price, atwhich point the auction ends.

• First-price sealed-bid auction. Each bidder privately sends a bid to the auction-eer (for example, in a sealed envelope). The highest bidder wins and pays herbid.

• Second-price sealed-bid (aka. Vickrey) auction. This auction is identical to thefirst-price sealed-bid auction, except the highest bidder (who still wins) pays thebid of the second-highest bidder.

A common objective among all of these auctions is to allocate the item to the bid-der who values the item the most (that is, to allocate the item efficiently). The differ-ences among them mostly reflect other issues, such as the following. One advantageof the English, Japanese, and Dutch auctions is that, depending on what happens in

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the auction, a bidder may not have to invest the effort necessary to determine her ex-act valuation for the item. For example, in an English or Japanese auction, once thesecond-to-last bidder drops out, the winner no longer has to think about how muchlonger she would have stayed in the auction; and in a Dutch auction, once someoneelse claims the item, the remaining bidders no longer have to think about when theywould have claimed the item. This is closely related to preference elicitation, whichwe will discuss in a later section. An advantage of the second-price sealed-bid auctionover the first-price sealed-bid auction is the following. In the first-price auction, bid-ders will try to only slightly outbid their competitors, to pay as little as possible. If thebidder with the highest valuation for the item underestimates the other bids, this canresult in another bidder winning the item, so that the allocation is inefficient. In thesecond-price sealed-bid auction, however, there is no reason to try to only slightly out-bid the next bidder; in fact, it can be shown that it is optimal to bid one’s true valuationfor the item. We will discuss issues such as these in a later section.

The main point, however, is that the various auctions agree, in some sense, onthe objective of efficient allocation, and the differences merely reflect other issues.There are other auctions that have a different objective, such as revenue-maximizingauctions [83], which do not always allocate the item efficiently; or even auctions thattry to minimize revenue (although most of these still allocate the item efficiently) [10,93, 20, 64, 80, 3, 63]. Nevertheless, in combinatorial auctions, typically the mainobjective is to allocate efficiently (though there is some work on revenue-maximizingcombinatorial auctions as well [9, 5, 38, 76, 77]). That is, it is assumed that everybidder i has a valuation function vi : 2I → R, where vi(S) is bidder i’s value for thesubset S of the items I , and the goal is to allocate to the bidders nonoverlapping bundlesSi to maximize

∑i vi(Si). As in the single-item auction case, there are distinctions

among combinatorial auctions in terms of the temporal aspects of the auction as wellas the payments to be made by the bidders, but these distinctions are again driven byother considerations, which we will discuss in later sections.1

2.2 VotingIn a typical voting setting, there is a set of alternatives or candidates, C, and each agent(voter) i has preferences i over these m alternatives, with a i b indicating that iweakly prefers a over b—that is, i either strictly prefers a to b, or is indifferent betweenthe two. Usually, for convenience, it is assumed that all preferences are strict, so thatwe simply use i. (Recent work has begun to extend social choice theory to settingswhere there is incompleteness/incomparability in the agents’ preferences [91, 98, 92].)Based on the preferences that the voters report, one alternative is chosen as the winner.(Sometimes, the output is a complete (aggregate) ranking of the alternatives; generally,a rule can be used to determine either just a winner or an aggregate ranking.) Some

1There are also variants of combinatorial auctions, such as combinatorial reverse auctions, where theauctioneer seeks to buy certain items and the bidders offer to sell bundles of these items at various prices;and combinatorial exchanges, where agents can be both buyers and sellers [105]. In another variant calledmixed multi-unit combinatorial auctions, agents can also bid to transform one set of goods into another set ofgoods [21]. While these variants are very important, they resemble regular combinatorial (forward) auctionsin terms of the issues discussed in this article, so we will not consider them in the remainder.

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example rules are given below. For each rule that gives points to alternatives, thealternative with the highest score is the winner. Generally, some tie-breaking schemeis required.

• Plurality. An alternative receives a point every time it is ranked first.

• Borda. An alternative receives m− 1 points every time it is ranked first, m− 2every time it is ranked second, ..., and 0 every time it is ranked last.

• Copeland. We conduct a pairwise election between every pair of alternatives:the winner of the pairwise election is the alternative that is ranked higher bymore voters. An alternative receives 2 points for every pairwise win, 1 point forevery pairwise tie, and 0 points for every pairwise loss.

• Bucklin. If there is an alternative that is ranked first by more than half the voters,that alternative wins; otherwise, if there is an alternative that is ranked first orsecond by more than half the voters, that alternative wins; etc.

• STV. The alternative that is ranked first the fewest times is removed from everyvote. (Votes that had this alternative ranked first will now have another alternativeranked first.) This is repeated until one alternative remains.

• Slater. We choose an aggregate ranking of the alternatives that is consistentwith the outcomes of as many pairwise elections as possible. (If a defeats b intheir pairwise election, then ranking a ahead of b would be consistent with theoutcome of this pairwise election, whereas ranking b ahead of a would not be.)

• Kemeny. We choose an aggregate ranking of the alternatives that has as fewdisagreements with votes as possible (a disagreement occurs when a vote ranksa above b, but the aggregate ranking ranks b above a; the disagreements aresummed over all votes and all pairs of alternatives).

In some sense, the differences among these rules are more fundamental than thedifferences among (combinatorial) auctions. Most auctions agree on the objective ofefficient allocation, and the differences among them are due more to other considera-tions, such as incremental preference revelation and strategic bidding. In contrast, ina voting setting, it is not even clear what objective we should be pursuing. Certainlywe can state a vague objective such as “the winner should be high in the preferences ofmany voters and low in the preferences of few,” but there are many different interpre-tations of this, and, in some sense, each of the voting rules above corresponds to onesuch interpretation.

One natural approach to identifying the “optimal” voting rule is to specify someaxioms that such a rule should satisfy. Unfortunately, there are several impossibilityresults that show that no rule satisfies certain natural properties. For example, con-sider the independence of irrelevant alternatives (IIA) criterion, which states that, if wemodify the votes but do not change whether a is ahead of b in any vote, then in theaggregate ranking, there also should be no change in whether a is ahead of b. Whilethis is a natural criterion, Arrow’s impossibility theorem [6] states that with 3 or more

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alternatives, any rule that satisfies IIA must either be Pareto inefficient, which meansthat it sometimes ranks b ahead of a even though all voters rank a ahead of b, or dic-tatorial, which means that the rule simply copies a fixed voter’s ranking. There aresome combinations of other axioms that are satisfied by exactly one rule—for exam-ple, Young and Levenglick [131] give an axiomatic characterization of the Kemenyrule. However, such axiomatic arguments have so far not succeeded in building broadconsensus on what the optimal rule is.

Another approach that has been pursued to decide on the optimal voting rule is thefollowing. Imagine that there exists a “correct” outcome (winner or ranking), which wecannot directly observe; but every voter’s preferences constitute a noisy observation ofthis correct outcome. Given a noise model (a conditional probability distribution overthe preferences given the correct outcome), it would make sense to choose the outcomethat maximizes the likelihood of the observed preferences. This approach was alreadypursued by the early social choice theorist Condorcet [46], who proposed one particu-lar noise model. In this noise model, each voter ranks each pair of alternatives correctlywith some constant probability p > 1/2, independently. (This can lead to cyclic pref-erences, but that does not affect the maximum likelihood approach.) Condorcet solvedfor the maximum likelihood estimator rule for the cases of 2 and 3 alternatives. Twocenturies later, Young showed that the solution for general numbers of alternatives co-incides with the Kemeny rule [130]. Unfortunately, this argument for the Kemeny ruleis convincing only to the extent that one believes that Condorcet’s noise model is thecorrect one. In fact, more recent results imply that for many (but not all) of the commonvoting rules, there is a noise model such that that rule becomes the solution [39, 35](another paper [111] deals with the case of the Borda rule specifically). Another re-cent paper [51] explores the relationship between this maximum likelihood frameworkand the distance rationalizability framework, where we find the closest “consensus”election that has a clear winner, for some definition of consensus and some distancefunction, and choose its winner.

It thus appears that for the foreseeable future, no general agreement will emerge onwhich rule is optimal.2 Moreover, as we move towards settings in which there are somany alternatives that it is no longer feasible for a voter to provide a full ranking of allof them, new rules must be designed to address this. We will discuss this in more detailin the next section.

3 Expressing preferencesIn the previous section, we defined the structure of the agents’ preferences (valuationsfor bundles in combinatorial auctions, and rankings of alternatives in voting). In orderto choose an outcome based on the agents’ preferences, the agents will need to reportthem; and to do so, the agents need a language in which to express their preferences. Itis generally easy to create a straightforward language for doing so, but expressing one’s

2A notable exception is the case where there are only two alternatives: in this case, there are good reasonsto consider the majority rule (the alternative preferred by more voters wins) optimal—and indeed commonvoting rules generally coincide with the majority rule in the two-alternative case.

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preferences in such a language often requires exponential space. A good language forexpressing preferences will allow agents to specify “natural” preferences concisely.

3.1 Combinatorial auctionsIn a combinatorial auction, the straightforward way for a bidder to communicate herpreferences to the auctioneer is to simply provide a list of 2|I| − 1 values, one for eachnonempty bundle of items. Of course, this is impractical for anything other than verysmall numbers of items. Hence, it becomes important to have a good bidding lan-guage in which bidders can express their valuation functions concisely. Unfortunately,regardless of the language used, there will always be some valuation functions that re-quire exponential space to express, for purely information-theoretic reasons. However,a good bidding language allows bidders to express natural valuation functions con-cisely. This is analogous to, for example, Bayesian networks (for an introduction toBayesian networks, see Chapter 14 of [101]): any distribution can be represented usinga Bayesian network, but in general, doing so requires specifying an exponentially largenumber of probabilities. However, natural distributions display a significant amountof conditional independence, and such distributions can be expressed much more con-cisely using a Bayesian network.

So, the question becomes: which valuation functions are natural? One common as-sumption is that the bidder is single-minded, that is, there is a bundle Si and a constantki such that vi(S) = ki if Si ⊆ S, and vi(S) = 0 if Si 6⊆ S. That is, the bidder hasher heart set on a particular bundle; if she gets it, she receives a utility of ki (and anyadditional items that she receives will simply be thrown away), but if even one itemfrom Si is missing, the bundle becomes worthless to her. A single-minded valuationfunction is easy to express: in this case, a bid consists simply of a bundle Si and avalue ki. However, usually, bidders are interested in more than one bundle. The ORlanguage3 allows a bidder to bid on multiple bundles. An example bid in this languageis (a, 3) OR (b, c, 4) OR (c, d, 5). This bid indicates that if the bidder receivesthe bundle a, b, c, her value is 7 (because she receives the first two bundles in herbid); if she receives b, c, d, her utility is 5 (each of the last two bundles in her bid arecontained in the bundle she receives, but c can only be counted towards one of them,and the last bundle has the greater valuation). A disadvantage of the OR language isthat it cannot express every valuation function. For example, consider the valuationfunction v(a) = 1, v(b) = 1, v(a, b) = 1 (the bidder wants either item andhas no use for a second item). If the bidder bids (a, 1) OR (b, 1), this implies avaluation of 2 for the bundle a, b; on the other hand, if she does not include bothsingleton bundles in her bid, then the bid will not reflect her valuation for those.

An alternative language is the XOR language. At most one of the bundles inan XOR-bid can be counted: for example, the bid (a, 3) XOR (b, c, 4) XOR(c, d, 5) indicates that the bidder’s valuation for a, b, c is 4 (only one of the first twobundles can be counted). In this language, any valuation function can be expressed (ifnecessary, by XOR-ing all possible bundles together). One downside of the XOR lan-guage is that some natural valuation functions require exponential space to represent—

3For a detailed discussion of the OR and XOR languages, see [85].

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for example, the function v(S) = |S| (which can be concisely represented in the ORlanguage by listing each singleton bundle at a value of 1). Of course, nothing preventsus from using ORs and XORs simultaneously, to obtain the best of both worlds.

There are also bidding languages of a different flavor. For example, the bidder canspecify interactions among the items [23, 44]. If a bidder reports a valuation of 2 foritem a, a valuation of 3 for item b, and an interaction of −1 between a and b, thisimplies a valuation of 2 + 3− 1 = 4 for the bundle a, b.

3.2 VotingIn a voting setting, a straightforward way for a voter to communicate her preferences isto simply communicate the position of each alternative in her ranking. If there arem al-ternatives (and hencem different positions), specifying one position requires Θ(logm)bits, leading to a total space requirement of Θ(m logm) bits. In many settings, this isquite manageable, and because of this the problem of how preferences are representedoften does not receive much attention in voting.

Nevertheless, there are many important settings in which there are exponentiallymany alternatives, so that Θ(m logm) space is no longer manageable. For example,in some settings, the set of alternatives may be written as C = X1 ×X2 × . . . ×Xp.Here, eachXj corresponds to a separate issue on which we need to make a decision. Insuch a domain, we need to make use of a more sophisticated language for representingpreferences, such as a CP-net [18]. (A CP-net allows an agent to specify whether andhow her preferences on one issue depend on the values chosen for the other issues.)Preferences expressed in such a language do not always give enough information torecover the full ranking of all alternatives, so not all the standard rules can be appliedin such a setting. Some very recent work has been devoted to determining how winnersshould be chosen in such settings [74, 124, 125, 121, 122], but much more remains tobe done here.

4 Winner determinationOnce we have defined the rules according to which an outcome is chosen, as well asthe language in which preferences are reported, we have a well-defined computationalproblem of deciding what the outcome is given the reported preferences. In combina-torial auctions, this computational problem is usually called the winner determinationproblem. Since in voting, we are also determining a winner, we will use the same namefor the problem in that setting.

4.1 Combinatorial auctionsThe winner determination problem in a combinatorial auction is to determine the al-location that maximizes the total value, given the bids (expressed in some biddinglanguage). For simplicity, let us first assume that every bidder i is single-minded, andhence her bid can be represented as (Si, ki). The problem is then to select a subset Aof the bids to accept, to maximize

∑i∈A ki, under the constraint that Si ∩ Sj = ∅ for

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all i, j ∈ A, i 6= j. This problem is NP-hard [100], even to approximate [102, 133].There are various approaches to solving it nonetheless, including: modeling it as an in-teger program, using a search-based algorithm [103], or using a dynamic programmingalgorithm [100].

It turns out that any algorithm for the winner determination problem with single-minded bids can be extended to deal with bids that use ORs and XORs. For ORs,this is easy to see: as far as winner determination is concerned, there is no differencebetween receiving two single-minded bids (S1, k1) and (S2, k2), or receiving a singlebid (S1, k1) OR (S2, k2), because in either case the auctioneer can award both bundlesfor a value of k1 +k2 if and only if the bundles do not overlap. This argument does notwork for XORs, but there is a clever trick for converting an XOR into an OR [58, 84]:given a bid (S1, k1) XOR (S2, k2), create a dummy item d, and replace the bid by(S1 ∪ d, k1) OR (S2 ∪ d, k2). Now the two bundles cannot both be awarded tothe bidder, because they overlap. Because of this, most research on the combinatorialauction winner determination problem has focused on single-minded bids.

While the winner determination problem with single-minded bids is NP-hard ingeneral, it can become polynomial-time solvable if the bids lie in a restricted class [100,110, 90, 104, 33, 82, 61]. For example, it is polynomial-time solvable if each bid is onat most two items [100]. It is also polynomial-time solvable if, in addition to the bids,we are given a graph of bounded treewidth whose vertices are the items and whoseedges are such that every bid is on a connected set of vertices/items [33] (this resultcan be generalized further [61]).

For other bidding languages, the complexity of the winner determination prob-lem may be different. For example, for the language based on specifying interactionsamong items (described above), the winner determination problem is NP-hard evenwith only pairwise interactions [23, 44].

4.2 VotingIn settings where the number of alternativesm is not extremely large, so that each votercan communicate her complete ranking of all the alternatives, determining the winningalternative is computationally straightforward for most voting rules. For example, arule such as Borda requires nothing more than adding up the scores of the alternativesand comparing them. However, this is not the case for all voting rules: some of themare in fact NP-hard to execute [13, 67, 26, 50, 99, 1, 2, 27, 96, 19]. As an example,let us take the Slater rule, which requires finding a ranking that is inconsistent withthe outcomes of as few pairwise elections as possible. Determining the outcomes ofthe pairwise elections is easy. It is helpful to summarize this information in a directedgraph, in which the alternatives are the vertices and each edge points from the winner ofa pairwise election to the loser. For simplicity, let us assume that none of the pairwiseelections end up tied (for example, we can assume that the number of voters is odd),so that there is an edge between every pair of alternatives. (For every graph of thisform, there exists a set of votes that results in this graph [78].) If this graph is acyclic,then it already corresponds to a ranking. If not, then the Slater rule asks us to make thegraph acyclic, by reversing as few edges as possible. This problem turns out to be NP-hard [1, 2, 27, 22]. The Kemeny rule can similarly be interpreted in a graph-theoretic

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way: to do so, we use the same graph as for the Slater rule, except we add a weightto each edge (a, b) which is equal to the margin of a’s pairwise victory over b. Theproblem now becomes to reverse a set of edges of minimum total weight to make thegraph acyclic. It is possible to find votes so that all the weights become equal to eachother [78], so the Kemeny problem is at least as hard as the Slater problem, that is, NP-hard. Still, it is possible to solve reasonably-sized winner determination instances forthe Kemeny rule, using search-based techniques [45, 79] or integer programming [32].Fixed-parameter tractability results for computing Kemeny rankings have also beengiven [15]. The remaining rules introduced earlier in this article (the ones other thanSlater and Kemeny) can be executed in polynomial time.

5 Preference elicitationSo far, we have assumed that each agent reports her complete preferences all at once.While a good language for describing preferences can be very helpful in doing so, theagent still needs to determine her complete preferences before she can report them. Analternative approach is to use a sequential process, where one agent reports some in-formation about her preferences; then, based on that information, another agent reportssome information about his preferences, etc., until we know enough to determine theoutcome. This approach is known as preference elicitation, and it is usually guided bya central party, the elicitor, who determines who has to report which information nextby asking that agent a query about her preferences.

When we use preference elicitation, it is generally not necessary for each agent toreveal all of her preferences. Very often, once we have obtained some preference in-formation from the agents, other information becomes irrelevant. For example, if wehave already determined that one agent’s valuation for a particular bundle is not highenough to have any chance of winning, then there may not be any need to determinethe exact valuation. As another example, if we have already determined that one al-ternative has no chance of winning, then there may not be any need to determine itsexact position in a voter’s preferences. Making use of such observations can reducethe amount of information that the agents have to communicate. However, the benefithere is more than mere communicational convenience. Often, the greater benefit is thatthe agent does not even need to determine her preferences completely. A significantamount of deliberation effort can be required to determine even a single bit of prefer-ence information (for example, which of two alternatives is preferred), so it can be veryvaluable to realize that this information is not needed. An additional benefit is that theagents’ privacy is improved, in the sense that they release less of their preference infor-mation. It follows that preference elicitation can be extremely valuable even in settingswhere preferences can be described concisely—for example, voting over a small set ofalternatives.

5.1 Combinatorial auctionsIn combinatorial auctions, it is easy to see the potential benefits of preference elic-itation. Even when armed with the best bidding language, determining one’s exact

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valuation for every bundle can be overwhelming. It is therefore not surprising thatmuch of the work on preference elicitation has focused on combinatorial auctions.

In a sense, single-item auctions such as the English, Japanese, and Dutch auctionsare already doing some preference elicitation. At each stage, bidders reveal some in-formation, such as whether the current price is too high for them; and once we knowthe winning bidder, we end the auction, even if we do not yet know all bidders’ exactvaluations. Indeed, some of the approaches to preference elicitation in combinatorialauctions mimic these single-item auctions. Several ascending combinatorial auctionshave been proposed [89, 7, 88]. In every round of an ascending combinatorial auction,each bidder faces a price for each bundle, and needs to decide which bundle she wouldlike to buy under these prices. The prices start low, so that bidders will choose largebundles that overlap with each other. Each round, prices are raised, up to the point thatthe chosen bundles no longer overlap with each other. While ascending combinatorialauctions can, under certain conditions, result in the optimal allocation, they also facesome inherent limitations [17].

Ascending combinatorial auctions typically proceed in a very systematic fashion.It is also possible to take a more flexible preference elicitation approach, where theauctioneer/elicitor can ask any agent any query at any point. Common queries includethe demand query, where a bidder is asked which bundle she would prefer under certainprices (as in the ascending auctions), and the value query, which simply asks for thebidder’s valuation for a specific bundle.

One question that is often studied in preference elicitation is the following: if abidder’s valuation function is guaranteed to lie in a certain class of functions C, canwe determine the bidder’s valuation function exactly using only a polynomial numberof queries? Various positive results have been obtained along this line. For example, abidder’s valuation function can be elicited exactly with a number of value and demandqueries that is polynomial in the length of the valuation function’s representation in theXOR language [73]. Hence, valuations that admit a concise XOR-representation canbe elicited efficiently using such queries. However, this result only holds if the demandqueries can set prices on bundles (rather than just on individual items) [16]. Thereare numerous other results that show that various classes can(not) be elicited using apolynomial number of queries of a given type [132, 107, 44, 17, 72].

If there are no restrictions on the valuation functions, some negative results areknown. For example, it has been shown using techniques from communication com-plexity theory [71] that the winner determination problem in general requires exponen-tial communication [87]. This is true regardless of the types of query that are allowed.

5.2 VotingWhen agents are voting over a set of alternatives that is not extremely large, the poten-tial benefits of preference elicitation are less dramatic than in a combinatorial auction:even reporting one’s complete preferences requires only a polynomial amount of com-munication. Nevertheless, as we have already noted, determining these preferencesgenerally still requires a large amount of deliberation effort. Hence, there is still sig-nificant value in doing preference elicitation in such settings.

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As in the case of combinatorial auctions, there has been some work on determininghow many queries are needed to completely elicit a voter’s preferences, assuming thatthe preferences lie in a specific class. For instance, sometimes there is a natural order<on the alternatives—for example, some candidates may be more “right-wing” than oth-ers. A voter’s preferences i are said to be single-peaked with respect to this order if,whenever a < b < c or c < b < a (where a is i’s most-preferred alternative), we havethat b i c. That is, the voter prefers alternatives that are closer to her most-preferredalternative. It has been shown that single-peaked preferences can be elicited using alinear number of comparison queries (which ask the voter which of two alternatives ispreferred), if either the order< is known, or at least one other vote that is single-peakedwith respect to < is known [30].

Another important topic is the communication complexity of executing variousrules—that is, for a given rule, what is the minimum number of bits that needs tobe communicated by the voters (in the worst case) to determine the winner?4 It shouldbe noted that here, there is no constraint on the types of query that are used. Someof the rules, such as Borda and Copeland, turn out to require Ω(nm logm) communi-cation in the worst case—that is, in the worst case, the communication requirementsare roughly as bad as having everyone reveal all of their preferences. For other rules,however, we can get away with much less communication: for example, STV requiresonly O(n(logm)2) (using the straightforward protocol where everyone reports theirmost-preferred alternative first; then, when an alternative is eliminated, all voters whoranked that alternative first communicate their next-most preferred alternative, etc.),and Bucklin requires only O(nm) (using a more sophisticated protocol based on bi-nary search) [40].

Another key computational problem for preference elicitation is the following [36,70, 92, 114, 117]: given some of the preferences of some of the voters, do we alreadyknow which alternative is the winner? If we can solve this problem, then we will knowwhen we can stop eliciting preferences and declare the winner. This problem is ofteneasy, but sometimes it is NP-hard, for example for STV [36].

A final direction is to optimize the elicitation process: given a prior distribution overthe voters’ preferences, plan the elicitation process to minimize the expected numberof voters whose preferences we elicit before we know the winner. For most rules, thisproblem is NP-hard even if we are completely certain beforehand about how voters willvote (but we still need to elicit their preferences to prove that we are right) [36].

6 Strategic agents and mechanism designA final important issue is that of strategic bidding/voting. Depending on the rules forthe auction/election and the bids/votes submitted by the other agents, it may not bein an agent’s best interest to truthfully report her preferences. An agent that reportspreferences other than the preferences she truly has (with the aim of obtaining a betteroutcome for herself) is said to misreport or manipulate. If agents manipulate, this can

4A related but distinct notion is that of compilation complexity, where the goal is to summarize the votesof a subset of the voters using as few bits as possible, while retaining all the information needed to computethe winner once the remaining votes become known [24, 120].

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be detrimental to the quality of the outcome: even if we manage to solve the winnerdetermination problem to optimality with respect to the reported preferences, there isno guarantee that this results in an outcome that is good with respect to the true pref-erences, which is what we really care about. As we will see shortly, the approachesthat are taken to address this problem in combinatorial auctions and in voting are gen-erally different. This is because in combinatorial auctions, the payments can be usedto remove any incentive for the bidders to misreport their valuation functions (makingit optimal for each agent to report truthfully is the standard approach in mechanismdesign)—whereas in sufficiently general voting settings, any reasonable voting rulewill lead to incentives to misreport in some cases.

6.1 Combinatorial auctionsTo illustrate the basic idea, it is helpful to return to the single-item auction setting.We assume quasilinear utilities, that is, that bidder i’s utility for winning the item atprice πi is vi − πi, where vi is the bidder’s (true) valuation for the item; and her utilityfor not winning (and paying nothing) is 0. Let us first consider the first-price sealed-bid auction. It is easy to see that in this auction, it never makes sense for a bidder tobid her true valuation vi, because if she does so, then even if she wins, her utility isvi − vi = 0. Rather, a bidder needs to bid lower than her true valuation to have anychance of obtaining positive utility.

In contrast, let us consider the second-price sealed-bid (Vickrey) auction. To seehow bidder i should bid in this auction, let us first suppose that she knows all the otherbids. Then, she has two choices: either bid higher than the highest other bid, b∗, andobtain utility vi − b∗; or bid lower, and obtain utility 0. Clearly, she should do theformer if and only if vi > b∗. But this can be achieved simply by bidding truthfully(for which she does not even need to know the others’ bids). Thus, it is always optimalfor a bidder to reveal her true valuation in a second-price sealed-bid auction. That is,this auction is strategy-proof.

A natural question is whether it is possible to make combinatorial auctions strategy-proof as well by using the right payments. It turns out that this is indeed possible, byusing the Generalized Vickrey Auction (GVA) (which is a special case of the Clarkemechanism, which in turn is a member of the class of VCG mechanisms [112, 25,62]). The GVA works as follows: first, solve the winner determination problem, toobtain an optimal allocation S1, . . . , Sn, with total value V =

∑ni=1 vi(Si). Now,

to determine how much bidder i must pay, remove i from the auction, and solve thewinner determination problem again with the remaining bidders, obtaining a total valueof V−i (which is at most V ). Then, bidder i must pay V−i − (V − vi(Si)). It is notdifficult to show that the GVA is strategy-proof.

While the GVA appears to solve the problem of strategic bidding nicely, it still hasa number of drawbacks. First of all, if obtaining the efficient allocation of the itemsis not the objective, then the GVA may not be optimal. For example, there has beensignificant interest in finding a combinatorial auction mechanism that maximizes ex-pected revenue, although this turns out to be surprisingly difficult [9, 5, 38, 76, 77].Another problem is that the GVA is very vulnerable to multiple bidders colluding:while there is no incentive for a single bidder to misreport by herself, a group of bid-

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ders can make themselves better off by collectively misreporting [8, 41]. A biddercan also make herself better off by pretending to be multiple distinct bidders, whichis often possible in open, anonymous environments such as the Internet. That is,the GVA is not false-name-proof. Several combinatorial auction mechanisms havebeen proposed that are false-name-proof, but they come at the cost of reduced effi-ciency [128, 127, 129, 65, 69]. In fact, fairly weak conditions that preclude false-name-proofness have been given [97], and a recent paper shows that the efficiency lossfrom using false-name-proof combinatorial auction mechanisms (that satisfy some ap-parently minor assumptions) is necessarily severe [69]. (An alternative approach todealing with false-name bidding is to verify the identities of some of the bidders afterthe fact [28].) A final issue is that the winner determination problems must be solvedto optimality: if an approximately optimal solution is used, then the resulting modifiedGVA is in general no longer strategy-proof. A significant body of research has focusedon creating approximation algorithms for the winner determination problem that can becombined with payment schemes that make them strategy-proof [75, 4, 48, 86, 47, 81].

6.2 VotingWhile the GVA mechanism solves the problem of strategic bidding in combinatorialauctions (with some significant caveats), in voting settings no such solution exists. Thisis fundamentally due to the lack of payments in voting settings: the Clarke mechanism,of which the GVA is a special case, can also be applied to (imaginary) voting settingsin which the voters can be required to make payments. But, without payments, there isthe Gibbard-Satterthwaite impossibility theorem [59, 108], which states that with threeor more alternatives, every voting rule that is strategy-proof is either dictatorial, or issuch that there is an alternative that cannot win, regardless of the votes.5 One maywonder if one can do better using voting rules that use randomization in their choice ofthe winner; however, a later paper by Gibbard [60] characterizes the class of strategy-proof randomized voting rules completely, and the result is still mostly negative inthat all the rules in the class have some undesirable properties. As an aside, the classbecomes even smaller if we require false-name-proofness [29], though this is mitigatedif casting additional votes comes at a cost [113].

Still, something must be done to address the problem of manipulation. Many re-cent papers have pursued the approach of using computational hardness as a barrier tomanipulation. The idea here is the following: while for any reasonable rule, there areguaranteed to be instances where there exists a beneficial manipulation (that is, a wayof voting insincerely that makes the voter better off), the voter will not be able to makeuse of this if the manipulation is computationally too hard to discover. Indeed, there hasbeen a number of results showing that the problem of finding a beneficial manipulationis NP-hard (for some rules, for some definitions of the manipulation problem) [12, 11,37, 52, 66, 43, 96, 126, 123]. While this shows that there is indeed some computa-tional barrier to manipulation, the result is not as strong as we would like. Specifically,NP-hardness is a worst-case measure of hardness: it shows that (assuming P6=NP) any

5Recent work [53] considers situations where the voting language is restricted in such a way that it isnot possible for voters to vote truthfully, and in this context defines a notion of voting “sincerely” that is notsubject to such impossibility results.

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algorithm for finding manipulations is going to require more than polynomial time onsome instances, but it could still be the case that the algorithm rapidly identifies a ben-eficial manipulation on most instances. It would be more satisfactory to have a resultthat shows that most instances are hard to manipulate. Unfortunately, recently, variousnegative results have been obtained that suggest that it is usually easy to find a manip-ulation for reasonable rules [42, 94, 95, 57, 119, 118, 49, 134, 116, 115, 123]. Still,it seems that the book is not yet closed on using computational hardness as a barrierto manipulation. Incidentally, computational hardness has also been considered as abarrier against other types of undesirable behavior, for example by the chairperson ofthe election [14, 68, 54, 96, 34, 55, 56].

Another issue is that if we use preference elicitation for a voting rule that is notstrategy-proof, this may introduce additional opportunities for manipulation. This isbecause from the queries that a voter is asked, she can infer something about how theothers are voting; and she may wish to change her vote based on this information [36].This is an issue that does not occur if we use a strategy-proof mechanism (such as theGVA): in that case, answering the queries truthfully becomes an ex post equilibrium.That is, it is optimal for every agent to answer the queries truthfully, as long as theothers do so as well—regardless of what their preferences are.

7 ConclusionsIn this article, we considered several issues that are of importance both in combinatorialauctions and in voting settings, and compared the research on these issues across thetwo domains.

We first considered the rules for how outcomes are chosen in combinatorial auc-tions and in voting. Here we noticed a major difference. Most (though not all) com-binatorial auction designs agree on the objective of maximizing the efficiency of theallocation of the items, and the differences among the designs are primarily due toother factors, such as how bidders’ valuations are elicited and what incentives biddershave when they bid strategically. In contrast, in voting settings, it is generally notclear what the ideal objective is, and different voting rules can be said to correspond todifferent objectives.

We then considered the languages in which agents express their preferences. Incombinatorial auctions, there is a significant body of research on various bidding lan-guages and their strengths and weaknesses. In voting, there is not much research onlanguages yet; this is presumably primarily because most work so far has dealt withsettings where the number of alternatives is not extremely large, and in such settingsit does not require much space to express any ranking. However, this is not true inthe case of combinatorial alternative spaces, which can have exponential size. Somelanguages for representing preferences in combinatorial alternative spaces have beenproposed, and as the interest in such settings grows, undoubtedly so will the interest invoting languages.

Subsequently, we considered the winner determination problem. In combinato-rial auctions, the winner determination problem has received a lot of attention, in partbecause this is perhaps the first problem that needs to be solved to be able to run a com-

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binatorial auction. In voting, when the number of alternatives is not extremely large,the winner determination problem is quite easy for most rules. But there are some rulesfor which it is NP-hard, and various techniques have been proposed for executing theserules nonetheless. It seems that as the focus shifts to combinatorial alternative spaces,more difficult winner determination problems will emerge.

We then considered preference elicitation, that is, sequentially querying the agentsfor parts of their preferences until we know enough to determine the outcome. Again,this is a problem that has received more attention in combinatorial auctions, presumablybecause in a general combinatorial auction, a bidder must communicate exponentiallymany values, so any reduction in this communication is highly desirable. Neverthe-less, preference elicitation has also received some attention in voting settings. This isbecause even though specifying a ranking requires space polynomial in the number ofalternatives, it generally requires significant deliberation effort to decide on the pre-cise ranking, and preference elicitation can reduce the required deliberation effort. Yetagain, it seems that preference elicitation will receive more attention in settings withexponentially many alternatives.

Finally, we considered strategic agents, who will misreport their preferences if thisis to their benefit. Here there is a major difference between combinatorial auctions andvoting. In combinatorial auctions, it is possible to remove any incentive to misreport,using techniques from mechanism design. Unfortunately, in voting settings, mecha-nism design provides little more than negative results, and hence research has turned toother ways to prevent misreporting—specifically, making it computationally hard to doso beneficially. Both in combinatorial auctions and in voting settings, there is growinginterest in addressing new types of manipulation, such as participating in the mecha-nism multiple times using false identifiers. It appears that this type of manipulation ismore difficult to address in combinatorial auctions than misreporting, so perhaps thetechniques that will be designed to address this will be similar across the two domains.It is not clear that the approach of making manipulation computationally hard will gaintraction in the combinatorial auctions literature, especially given the recent negativeresults about making manipulation usually hard in voting settings; nevertheless, therehas already been some work on the hardness of various types of manipulation in com-binatorial auctions [106, 41, 109].

In summary, many of the same issues are at play in combinatorial auctions andvoting settings. Considering current trends in research in the two domains, it seemsthat the similarities will only grow. I speculate that future research on computationalaspects of voting will increasingly focus on exponentially-sized alternative spaces; itseems that here in particular, considering techniques from the combinatorial auctionsliterature will be useful. Also, I speculate that future research on combinatorial auc-tions will increasingly consider settings where manipulation cannot reasonably be en-tirely disincentivized via mechanism design techniques; since we already face such asituation in voting settings due to results such as the Gibbard-Satterthwaite impossibil-ity theorem, drawing connections seems helpful. Rather than developing two separate(though possibly analogous) theories for the two different domains, it would be desir-able to develop a single coherent theory that can be applied to both domains. Hence, itappears that researchers in each domain would benefit from keeping a close eye on theother domain.

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